Functional equations and distribution functions
type: Item
instance of: scholarly article
title: Functional equations and distribution functions
zbMATH Open document ID: 0837.39007
DOI: 10.1007/BF03323043
author: Roland Girgensohn
author: Jonathan M. Borwein
published in: Results in Mathematics
publication date: 1996-05-13T00:00:00Z
review text: Let \(a \in (0,1)\), \(N \in \mathbb{N} \backslash \{1\}\) and \(- 1 = \beta_0 \leq \beta_1 \leq \dots \leq \beta_{N - 1} = 1\). Then the functional equation \[ f(x) = {1 \over N} \sum^{N - 1}_{k = 0} f \left( {x - \beta_k \over a} \right) \] has a unique bounded solution \(f : \mathbb{R} \to \mathbb{R}\) vanishing on \((- \infty, -1/(1 - a))\) and taking the value 1 on \((1/(1 - a), + \infty)\). This unique solution \(f\) is continuous and increasing. It is strictly increasing on \([- 1/(1 - a), 1/(1 - a)]\) iff \(2a/(1 - a) \geq \max \{\beta_k - \beta_{k - 1} : 1 \leq k \leq N - 1\}\). If \(2a/(1 - a) < \min \{\beta_k - \beta_{k - 1} : 1 \leq k \leq N - 1\}\) then \(f\) is singular. It is also singular in the case where \(\beta_k = - 1 + 2k/(N - 1)\) for \(k \in \{0, \dots, N - 1\}\) and \(a\) is the inverse of a nonrational Pisot number. If \(\beta_k = - 1 + 2k/(N - 1)\) for \(k \in \{0, \dots, N - 1\}\) and \(a = N^{- 1/p}\) with an integer \(p\), then \(f\) is absolutely continuous, has a \((p - 1)st\) derivative which is a continuous and piecewise linear function (and a formula for \(f\) is given via convolutions). An interesting connection with the Schilling equation \[ 4q \varphi (qx) = \varphi (x - 1) + \varphi (x + 1) + 2 \varphi (x), \] where \(q \in (0,1)\) is a fixed parameter, is also pointed out.
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 785165
zbMATH Keywords: distribution functions
zbMATH Keywords: Bernoulli distributions
zbMATH Keywords: singular and absolutely continuous functions
zbMATH Keywords: Vieta's product
zbMATH Keywords: Pisot numbers
zbMATH Keywords: functional equation
zbMATH Keywords: bounded solution
zbMATH Keywords: Schilling equation
reviewed by: Karol Baron
MaRDI profile type: Publication
cites work: Q4323683
cites work: On a Family of Symmetric Bernoulli Convolutions
cites work: On the Smoothness Properties of a Family of Bernoulli Convolutions
cites work: Arithmetic Properties of Bernoulli Convolutions
cites work: Q3218611
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cites work: On bounded solutions of a problem of R. Schilling
cites work: One-parameter system of functional equations
description: scientific article; zbMATH DE number 785165